Random signals are stochastic processes that model unpredictable electrical phenomena and noise in communication systems.

Key Applications in EE:
- Communication Systems: Channel modeling, noise analysis
- Signal Processing: Filtering, detection, estimation theory
- Control Systems: System identification, adaptive control
- Circuit Analysis: Thermal noise, shot noise modeling
- Antenna Systems: Random electromagnetic fields
- Digital Signal Processing: Quantization noise, aliasing effects

Why Stochastic Processes Matter:
- Real-world signals contain unpredictable components
- Noise is fundamental limitation in all electronic systems
- Statistical models enable optimal system design
- Foundation for modern wireless communications

Stochastic Process Definition: A random signal \(X(t)\) is a stochastic process where each \(t\) represents time and \(X(t)\) is a random variable representing signal amplitude.

Key Statistical Characterizations:
- Mean (DC Component): \(\mu_X(t) = E[X(t)]\)
- Autocorrelation Function: \(R_X(t_1,t_2) = E[X(t_1)X(t_2)]\)
- Power Spectral Density: \(S_X(\omega) = \mathcal{F}[R_X(\tau)]\)

For WSS Signals:
\[R_X(\tau) = E[X(t)X(t+\tau)]\\ S_X(\omega) = \int_{-\infty}^{\infty} R_X(\tau) e^{-j\omega\tau} d\tau\]

The total average power is: \(P = R_X(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_X(\omega) d\omega\)

White Noise Process:
- Autocorrelation: \(R_n(\tau) = \frac{N_0}{2}\delta(\tau)\)
- Power Spectral Density: \(S_n(\omega) = \frac{N_0}{2}\) (constant)
- Physical Origin: Thermal noise in resistors, shot noise in diodes

Colored Noise (Filtered White Noise): When white noise passes through LTI system with impulse response \(h(t)\):
\[S_y(\omega) = |H(\omega)|^2 S_n(\omega)\]

Thermal Noise in Resistors: Where \(k\) = Boltzmann constant, \(T\) = temperature, \(R\) = resistance, \(B\) = bandwidth
\[\overline{v_n^2} = 4kTRB\]

##    time transmitted       noise   received
## 1 0.000   -1.000000  0.22763554 -0.7723645
## 2 0.001   -0.809017 -0.55425783 -1.3632748
## 3 0.002   -0.309017  1.11420761  0.8051906
## 4 0.003    0.309017  0.45459846  0.7636155
## 5 0.004    0.809017  0.06347036  0.8724874
## 6 0.005    1.000000  0.19555091  1.1955509
## 7 0.006    0.809017  0.48032973  1.2893467
## 8 0.007    0.309017  0.06352144  0.3725384

System Model: \(r(t) = s(t) + n(t)\)

- Signal-to-Noise Ratio (SNR): \(\frac{P_s}{P_n} = \frac{\text{var}(s)}{\text{var}(n)}\)

## 
## Signal-to-Noise Ratio: 0.11 dB